Quotations and References

Carl Friedrich Gauss: Gauss (1777 – 1855) was a German mathematician who made outstanding contributions to a wide range of areas including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, mechanics, electrostatics, astronomy, matrix theory, and optics. He is ranked as one of the greatest mathematicians of all time. There are numerous biographies of Gauss including Gauss: A Biographical Study by Walter Kaufmann Bühler and Carl Friedrich Gauss: Titan of Science by G. Waldo Dunnington.

The following quote by Gauss was contained in a letter to Sophie Germain (30 April 1807) and ends with the praise that “she must without doubt have the noblest courage, quite extraordinary talents and superior genius.”

The enchanting charms of this sublime science [mathematics] reveal themselves in all their beauty only to those who have the courage to go deeply into it.

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Sofia Vasilyevna Kovalevskaya: Kovalevskaya (1850–1891) was the first major Russian female mathematician, and was responsible for important original contributions to analysis, differential equations and mechanics. Despite her exceptional ability, partly because she was a woman and partly because of her social views, in her early years she had to overcome a great deal of prejudice. When she moved to Sweden she referred to herself as Sonya.

In 1888 Kovalevskaya won the Prix Bordin from the French Academie Royale des Sciences for research now called the Kovelevskaya top. This research examined how Saturn’s rings rotated. She also won a prize from the Swedish Academy of Sciences in 1889, and that same year was appointed to a chair at Stockholm University – the first woman appointed to a chair at a modern European university. She was also elected to the Russian Academy of Sciences as a member.

The following quote by Kovalevskaya is quoted in Women in Mathematics by Lynn M. Osen, The MIT Press, 1974, p. 136.

Many who have never had an opportunity of knowing any more about mathematics confound it with arithmetic, and consider it an arid science. In reality, however, it is a science which requires a great amount of imagination, and one of the leading mathematicians of our century states the case quite correctly when he says that it is impossible to be a mathematician without being a poet in soul.

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David Hilbert: Hilbert (1862 – 1943), a German mathematician, is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis. Hilbert is known as one of the founders of proof theory and mathematical logic.

He is perhaps most well-known to the academic community and even the general public for the presentation he gave in 1900 of a collection of problems that set the course for much of the mathematical research of the 20th century. The list of twenty-three problems were all unsolved at the time and some are unsolved even today.

In a lecture Hilbert gave in 1924 he introduced the following amusing story now referred to as the story of Hilbert’s Hotel. This hotel had a (countably) infinite number of rooms. The paradoxical feature of this hotel is that even when it was full, it always had room for another guest. The manager would simply move the guest in room #1 to room #2. Then the guest in room #2 he would move to room #3, and so on. Thereupon he would place the new guest in the now empty room #1.

The following quote by Hilbert is given in To Infinity and Beyond (Boston, 1987) by E. Maor.

No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.

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Albert Einstein: Einstein (1879 – 1955) was a German-born theoretical physicist who developed the general theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics). He also made significant contributions to the philosophy of science. He received the 1921 Nobel Prize in Physics. When Adolf Hitler came to power in 1933, Einstein was visiting the United States and, being Jewish, did not go back to Germany, where he had been a professor at the Berlin Academy of Sciences. He settled in the U.S., becoming an American citizen in 1940. Einstein published more than 300 scientific papers along with over 150 non-scientific works.

As one example of his work, between 1907 and 1915 Einstein developed the theory of general relativity whereby the observed gravitational attraction between masses results from the warping of space and time by those masses.

In various articles Einstein talks about the role played by “thought experiments” in helping him formulate some of his ideas. For example, he wrote that when he was 16 years old he tried to imagine what it would be like to chase a light beam travelling at the speed of light. He explained that this “experiment” was part of the genesis of his theory of special relativity.

His general writing on a wide range of topics such as pacifism, government, philosophy of science, creativity are well known and frequently quoted. The following quotation is taken from the English translation of an address he gave on 27 January 1921 at the Prussian Academy of Sciences in Berlin titled Geometry and Experience. The lecture was published by Methuen & Co. Ltd, London, in 1922.

How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?

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Eugene Wigner: Wigner (1902 – 1995) was a Hungarian American theoretical physicist and mathematician. He received half of the Nobel Prize in Physics in 1963 “for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles”.

In his later years he became increasingly interested in the philosophy of knowledge and he wrote an important paper titled The Unreasonable Effectiveness of Mathematics in the Natural Sciences. (Communications on Pure and Applied Mathematics 13: 1–14. 1960) The following quote is taken for this paper.

It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of laws of nature and of the human mind’s capacity to divine them.

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Maharishi Mahesh Yogi: Maharishi is well-known for introducing Transcendental Meditation® to the West. He is also a Vedic scholar who has written many books including Maharishi Mahesh Yogi on The Bhagavad Gita. As well, Maharishi is responsible for revitalisation of many areas of ancient Vedic knowledge including Ayurveda (traditional medicine) and Sthapatya Veda (Vedic architecture). He is the founder of Maharishi University of Management as well as hundreds of schools throughout the world using Conscious-Based Education.

The following quote by Maharishi is contained in his book Vedic Knowledge for Everyone (p. 67-68).

Consciousness is fundamental to life. All speech, action, and behavior are fluctuations of consciousness. The whole universe is the expression of consciousness. The reality of the universe is one unbounded ocean of consciousness in motion. Since consciousness is the most basic element of everyone’s life, knowledge of consciousness is the most basic requirement for everyone to exist consciously and intelligently and enjoy full, unbounded potential of life, with maximum success in all fields of personal and professional life.

Another quote by Maharishi from the same book particularly relevant to this Journal is:

Consciousness is that which is conscious of itself. Being conscious of itself, consciousness is the knower of itself. Being the knower of itself, consciousness is both the knower and the known. Being both the knower and the known, consciousness is also the process of knowing. Thus consciousness has three qualities within its self-referral singularity—the qualities of knower, knowing and known—the three qualities of ‘subject’ (knower), ‘object’ (known), and the relationship between the subject and object (process of Knowing). (p.53)

© 2015 International Journal of Mathematics and Consciousness.